On Hamiltonian Graphs Arising From Spatial Orders of Chromosomes of Even Number

Abstract

Abstract. According to Bennett's model of cytogenetics the spatial order in haploid chromosome complements is based on a similarity relation which gives rise to a multigraph G which is the edge-disjoint union of two of its subgraphs G\ and G2. For even chromosome numbers n Bennett's model postulates that the order of the n chromosomes is given by a Hamiltonian circuit of G alternating in the edges of Gi and G2. However, such a Hamiltonian circuit does not always exist. We impose a weak condition on the similarity relation and prove that under this condition the assumed Hamiltonian circuit does exist for all even chromosome numbers n < 50, which settles the case for all biological relevant species with n pairs of chromosomes. Moreover we study the structure of the graph G in respect to cycle decompositions and possible generalizations of our results. IntroductionAn individual chromosome in a cell of a eucaryotic organism consists of a short arm and of a long arm which are linked at the so-called centromere. During a certain stage of cell division, called metaphase, the centromeres of the n chromosomes of a haploid complement have approximately the form of a plane regular n-gon in which the arms of the chromosomes axe stretched to the outside. According to Bennett's model ([1], [2], [3]) the arms are arranged in such a way that always two short arms and two long arms are adjacent with one possible exception, if n is odd. Moreover, adjacent arms are assumed to be of "most similar size". However, preparing the cell for the electron microscope often destroys the assumed order, but usually allows to identify the individual arms of the chromosomes and to measure their lengths. In order to understand the mechanism of pairing (in areas like medicine, plant breeding, and genetic engineering) it may be important to know the original order of the chromosomes. To solve the problem of reconstructing the order of the chromosomes on the basis of arm lengths, Bennett recommended a procedure which in terms of mathematics runs as follows: Let s" and l" denote the short arm and the long arm of chromosome v, u = 1,2,..., n. All short arms s v , the lengths of which are assumed to be pairwise different, and all long arms l v which are also assumed to be pairwise different in length, are separately ranked in descending order of size. This ranking gives rise to two chains of indices S and L referring to the short and long arms, respectively. Without loss of generality we assume that S = (1,2,..., n) (which means that we will have to change the numbers usually assigned to the chromosomes in biology), so that L = (7rl,7r2,... ,-rm) for an appropriate permutation 7r € S n (symmetric group on n letters). Hence any measurement of the arm-lengths corresponds to a unique permutation 7T € S n -Next we interpret the notion "most similar size" of arms: For p, q € {1,2,..., n},p / <7 we say that two short arms s p and s q are k-similar, if \p -q\ < k, and analogously we call two long arms l np , l nq fc-similar, if \p -q\ < k. li k = 1, the arms ...